Alexandrov groupoids and the nuclear dimension of twisted groupoid $\mathrm{C}^*$-algebras

Abstract

We consider a twist E over an étale groupoid G. When G is principal, we prove that the nuclear dimension of the reduced twisted groupoid C * -algebra is bounded by a number depending on the dynamic asymptotic dimension of G and the topological covering dimension of its unit space. This generalizes an analogous theorem by Guentner, Willett, and Yu for the C * -algebra of G. Our proof uses a reduction to the unital case where G has compact unit space, via a construction of "groupoid unitizations" G and E of G and E such that E is a twist over G. The construction of G is for r-discrete (hence étale) groupoids G which are not necessarily principal. When G is étale, the dynamic asymptotic dimension of G and G coincide. We show that the minimal unitizations of the full and reduced twisted groupoid C * -algebras of the twist over G are isomorphic to the twisted groupoid C * -algebras of the twist over G. We apply our result about the nuclear dimension of the twisted groupoid C * -algebra to obtain a similar bound on the nuclear dimension of the C * -algebra of an étale groupoid with closed orbits and abelian stability subgroups that vary continuously.